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Diophantine representation of the decimal expansions of \(e\) and \(\pi \). (English) Zbl 0984.11007
Summary: Let \(\alpha \in \{\operatorname {e},\pi \}\), \(\alpha =[\alpha]+\sum _{\kappa =1}^\infty \alpha _\kappa(\beta)\cdot \beta ^{-\kappa}\) (where \(\beta \in \mathbb N\setminus \{1\}\) and \(\alpha _\kappa(\beta)\in \{0,1,\dots ,\beta -1\}\)) and \(\zeta \in \{0,1,\dots ,\beta -1\}\). We describe short Diophantine representations for the predicate \(\alpha _\kappa(\beta)=\zeta \). The proofs use methods that were developed for the solution of Hilbert’s Tenth Problem.
MSC:
11A63 Radix representation; digital problems
11U05 Decidability (number-theoretic aspects)
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References:
[1] BAILEY D. H.-BORWEIN J. M.-BORWEIN P. B.-PLOUFFE S.: The quest for Pi. Math. Intell. 19 (1997), 50-57. · Zbl 0878.11002
[2] BAXA C.: A note on Diophantine representations. Amer. Math. Monthly 100 (1993), 138-143. · Zbl 0805.11085
[3] DAVIS M.: Hilber\?s Tenth Problem is unsolvable. Amer. Math. Monthly 80 (1973), 233-269 · Zbl 0277.02008
[4] DAVIS M.-MATIJASEVIČ, YU. V.-ROBINSON J.: Hilber\?s Tenth Problem. Diophantine equations: Positive aspects of a negative solution. Mathematical Developments Arising from Hilbert Problems (F. E. Browder, Amer. Math. Soc, Providence, RI, 1976.
[5] DAVIS M.-PUTNAM H.-ROBINSON J.: The decision problem for exponential Diphantine equations. Ann. Mat\?. 74 (1961), 425-436. · Zbl 0111.01003
[6] JONES J. P.: Diophantine representation of Mersejine and Fermat primes. Acta Arith. 35 (1979), 209-221. · Zbl 0341.02036
[7] JONES J. P.: Universal Diophantine equation. J. Symb. Logic 47 (1982), 549-571. · Zbl 0492.03018
[8] JONES J. P.-MATIJASEVIČ, JU. V.: A new representation for the symmetric binomial coefficient and its applications. Ann. Sci. Math. Québec 6 (1982), 81-97. · Zbl 0499.03028
[9] JONES J. P.-MATIJASEVIČ, YU. V.: Proof of recursive unsolvability of Hilber\?s Tenth Problem. Amer. Math. Monthly 98 (1991), 689-709. · Zbl 0746.03006
[10] JONES J. P.-SATO D.-WADA H.-WIENS D.: Diophantine representation of the set of prime numbers. Amer. Math. Monthly 83 (1976), 449-464. · Zbl 0336.02037
[11] MANIN, YU. I.: A Course in Mathematical Logic. Springer, New York, 1977. · Zbl 0383.03002
[12] MATIJASEVIČ, JU. V.: Enumerable sets are Diophantine. Soviet Math. Doklady 11 (1970), 354-358. · Zbl 0212.33401
[13] MATIJASEVIČ, JU. V.: Diophantine representation of the set of prime numbers. Soviet Math. Doklady 12 (1971), 249-254. · Zbl 0222.10018
[14] MATIYASEVICH, YU. V.: Hilber\?s Tenth Problem. MIT Press, Cambridge-Massachusetts, 1993.
[15] PUTNAM H.: An unsolvable problem in number theory. J. Symb. Logic 25 (1960), 220-232. · Zbl 0108.00701
[16] SMORYŃSKI C.: Logical Number Theory I. Springer, Berlin, 1991. · Zbl 0759.03002
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